Termination of spiral wave breakup in a Fitzhugh-Nagumo model via short and long duration stimuli

Rotating spiral waves have been observed in a variety of nonlinear biological and physical systems. Spiral waves are found in excitable and oscillatory systems and can be stationary, meander, or even degenerate into multiple unstable rotating waves (a process called "spiral wave breakup"). In the heart, spiral wave breakup is thought to be the underlying mechanism of cardiac fibrillation. The spatiotemporal complexity of multiple unstable spiral waves is difficult to control or terminate. Here, the mechanisms of the termination of spiral wave breakup in response to global stimulation are investigated. A modified Fitzhugh-Nagumo model was used to represent cellular kinetics to study the role of the fast (activation) and slow (recovery) variables. This simplified model allows a theoretical analysis of the termination of spiral wave breakup via both short and long duration pulses. Simulations were carried out in both two-dimensional sheets and in a three-dimensional geometry of the heart ventricles. The short duration pulses affected only the fast variable and acted to reset wave propagation. Monophasic pulses excited tissue ahead of the wave front thus reducing the amount of excitable tissue. Biphasic shocks did the same, but they also acted to generate new wave fronts from the pre-existing wave tails by making some active regions excitable. Thus, if the short duration stimuli were strong enough, they acted to fill in excitable tissue via propagating wave fronts and terminated all activity. The long duration wave forms were selected such that they had a frequency spectrum similar to that of the pseudoelectrocardiograms recorded during fibrillation. These long duration wave forms affected both the recovery and activation variables, and the mechanism of unstable multiple spiral wave termination was different compared to the short duration wave forms. If the long duration stimuli were strong enough, they acted to alter the "state" (i.e., combination of fast and slow variables) of the tissue throughout 1.5 cycles, thus "conditioning" the tissue such that by the end of the stimuli almost no excitable tissue remained. The peak current, total energy, and average power of stimuli required to terminate spiral wave breakup were less for the long duration wave forms compared to the short duration wave forms. In addition, closed loop feedback via stimulation with a wave form that was the difference of the pseudoelectrocardiogram and a strongly periodic chaotic signal was successful at terminating spiral wave breakup. These results suggest that it may be possible to improve cardiac defibrillation efficacy by using long duration wave forms to affect recovery variables in the heart as opposed to the traditional brief duration wave forms that act only on the fast variables. (c) 2002 American Institute of Physics.     

Spontaneous spiral wave breakup caused by pinning to the tissue defect

The work presents a mechanism of spiral wave initiation due to the specific boundary conditions on a border of cardiac tissue defect. There are known scenarios when anatomical or functional defects in cardiac tissue may provoke the spiral wave origination, including unidirectional blockage while passing through the narrow gates, bent over critical curvature wave fronts, inhomogeneous recovery of the tissue, etc. We show a new scenario of spiral wave breakup on a small defect, which is unexcitable but permeable for ionic currents supporting the excitation wave. It was believed that such defects stabilize the rotating wave; however, as shown, instead of stabilizing it leads to the spiral breakup and subsequent multiplication of the rotating waves.     

具有早期后除极化现象的可激发系统中螺旋波破碎方式研究

在Greenberg-Hasting元胞自动机模型中引入了正常元胞和老化元胞,并规定只有老化元胞存在早期后除极化现象且早期后除极化可以激发其他元胞.在正常元胞和老化元胞均匀分布的情况下,研究了早期后除极化对螺旋波演化行为的影响,重点探讨了早期后除极化导致的螺旋波破碎方式.数值模拟结果表明:早期后除极化在比率约为26.4%的少数情况下不对螺旋波产生影响,在其他情况下则会对螺旋波产生各种影响,包括使螺旋波漫游、漂移、波臂发生形变以及导致螺旋波破碎和消失等.观察到早期后除极化通过传导障碍消失和通过转变为反靶波消失,早期后除极化导致螺旋波破碎有8种方式,包括非对称破缺导致的破碎、对称破缺导致的破碎、同时激发双波导致的破碎、非对称激发导致的破碎、整体传导障碍导致的破碎、整体快速破碎等.分析发现这些螺旋波破碎现象都与早期后除极化产生回火波有关,得到螺旋波破碎的总比率通常约为13.8%,但是在适当选取老化元胞密度和早期后除极化的激发下,螺旋波破碎比率可达到32.4%,这些结果与心律失常致死的统计结果基本一致,本文对产生这些现象的物理机理做了简要分析.     

Beyond the Kuramoto-Zel’dovich theory: Steadily rotating concave spiral waves and their relation to the echo phenomenon

In numerical experiments with the Fitzhugh-Nagumo set of reaction-diffusion equations describing two-dimensional excitable media, unusual solutions are found that correspond to a concave spiral wave steadily rotating round a circular obstacle in a finite-size medium. Such a wave arises in the region of parameters corresponding to the solitonlike regime (see text); it appears due to the interaction between the peripheral areas of a “seed” spiral wave with a convex front and the echo waves incoming from the outer boundaries of a medium. The solutions obtained are in contradiction with intuition and represent a numerical counterexample to the known theories that forbid steadily moving excitation waves with concave fronts. Nevertheless, a concave spiral wave is a stable object; being transformed to the usual spiral wave with a convex front by suppressing echo at the outer boundaries of the medium, it is again recovered upon restoring the echo conditions. In addition to the single-arm spiral concave wave, solutions are obtained that describe multiarm waves of this type; for this reason, the concave fronts of these waves are a coarse property.     

Patterns of spiral wave attenuation by low-frequency periodic planar fronts

There is evidence that spiral waves and their breakup underlie mechanisms related to a wide spectrum of phenomena ranging from spatially extended chemical reactions to fatal cardiac arrhythmias [A. T. Winfree, The Geometry of Biological Time (Springer-Verlag, New York, 2001); J. Schutze, O. Steinbock, and S. C. Muller, Nature 356, 45 (1992); S. Sawai, P. A. Thomason, and E. C. Cox, Nature 433, 323 (2005); L. Glass and M. C. Mackey, From Clocks to Chaos: The Rhythms of Life (Princeton University Press, Princeton, 1988); R. A. Gray et al., Science 270, 1222 (1995); F. X. Witkowski et al., Nature 392, 78 (1998)]. Once initiated, spiral waves cannot be suppressed by periodic planar fronts, since the domains of the spiral waves grow at the expense of the fronts [A. N. Zaikin and A. M. Zhabotinsky, Nature 225, 535 (1970); A. T. Stamp, G. V. Osipov, and J. J. Collins, Chaos 12, 931 (2002); I. Aranson, H. Levine, and L. Tsimring, Phys. Rev. Lett. 76, 1170 (1996); K. J. Lee, Phys. Rev. Lett. 79, 2907 (1997); F. Xie, Z. Qu, J. N. Weiss, and A. Garfinkel, Phys. Rev. E 59, 2203 (1999)]. Here, we show that introducing periodic planar waves with long excitation duration and a period longer than the rotational period of the spiral can lead to spiral attenuation. The attenuation is not due to spiral drift and occurs periodically over cycles of several fronts, forming a variety of complex spatiotemporal patterns, which fall into two distinct general classes. Further, we find that these attenuation patterns only occur at specific phases of the descending fronts relative to the rotational phase of the spiral. We demonstrate these dynamics of phase-dependent spiral attenuation by performing numerical simulations of wave propagation in the excitable medium of myocardial cells. The effect of phase-dependent spiral attenuation we observe can lead to a general approach to spiral control in physical and biological systems with relevance for medical applications.     

Multiple mechanisms of spiral wave breakup in a model of cardiac electrical activity

It has become widely accepted that the most dangerous cardiac arrhythmias are due to reentrant waves, i.e., electrical wave(s) that recirculate repeatedly throughout the tissue at a higher frequency than the waves produced by the heart's natural pacemaker (sinoatrial node). However, the complicated structure of cardiac tissue, as well as the complex ionic currents in the cell, have made it extremely difficult to pinpoint the detailed dynamics of these life-threatening reentrant arrhythmias. A simplified ionic model of the cardiac action potential (AP), which can be fitted to a wide variety of experimentally and numerically obtained mesoscopic characteristics of cardiac tissue such as AP shape and restitution of AP duration and conduction velocity, is used to explain many different mechanisms of spiral wave breakup which in principle can occur in cardiac tissue. Some, but not all, of these mechanisms have been observed before using other models; therefore, the purpose of this paper is to demonstrate them using just one framework model and to explain the different parameter regimes or physiological properties necessary for each mechanism (such as high or low excitability, corresponding to normal or ischemic tissue, spiral tip trajectory types, and tissue structures such as rotational anisotropy and periodic boundary conditions). Each mechanism is compared with data from other ionic models or experiments to illustrate that they are not model-specific phenomena. Movies showing all the breakup mechanisms are available at http://arrhythmia.hofstra.edu/breakup and at ftp://ftp.aip.org/epaps/chaos/E-CHAOEH-12-039203/ INDEX.html. The fact that many different breakup mechanisms exist has important implications for antiarrhythmic drug design and for comparisons of fibrillation experiments using different species, electromechanical uncoupling drugs, and initiation protocols. (c) 2002 American Institute of Physics.     

Resetting wave forms in dictyostelium territories

The mechanism by which spiral wave patterns appear in populations of Dictyostelium was probed experimentally by external chemical perturbation. Spiral waves, which often arise from the breakup of circular waves driven by pacemakers, typically entrain those pacemakers. We studied these processes by resetting the waves with a spatially uniform pulse of extrinsic cyclic AMP. A pattern of spirals reappeared if resetting was early in the signaling stage, but only targets emerged following late resetting, in a manner analogous to cardiac defibrillation. This supports recent hypotheses that wave pattern selection naturally occurs by slow temporal variation of the excitability of the cells.     

心肌组织机械形变的元胞自动机模拟

采用Greenberg-Hastings元胞自动机模型研究机械形变对心肌组织中螺旋波动力学行为的影响.数值模拟表明：对于规则网格下的稳定螺旋波,在生理性机械形变作用下,螺旋波发生漫游但不破碎;在病理性机械形变作用下,螺旋波会发生持续漫游、漫游后消失和破碎进入螺旋波湍流态三种变化.通过对比发现机械形变的振幅变化率对螺旋波的影响较大,而机械形变的角频率对螺旋波的影响较小.结合数值模拟,对心前区受到猛烈撞击会出现心颤致死及耐力运动员在发生心动过速后比一般人员更容易恢复正常进行解释.     

Dynamics of Spiral Waves Induced by Periodic Mechanical Deformation with Phase Difference

The dynamics of spiral waves under the influences of periodic mechanical deformation are studied. Here, the mechanical deformation propagating along the medium with phase differences are considered. It is found that weak mechanical deformation may lead to resonant drift of spiral waves. The drift direction and velocity can be changed by the wave length of the deformation. Strong mechanical deformation may result in breakup of spiral waves. The characteristics of breakup are discussed. The critical amplitudes are determined by two factors, i.e. the wave length and frequency of the periodic mechanical deformation. When the wave length of mechanical deformation is comparable to the spiral wave, simulation shows that the critical amplitude is substantially increased. As the frequency of the mechanical deformation is around 1.5 times of the spiral wave, the critical amplitudes are minimal.     

时空调制对可激发介质螺旋波波头动力学行为影响及控制研究

本文首先研究了时空调制对可激发介质中周期螺旋波波头动力学行为的影响. 随着时空调制的增大, 螺旋波经历了周期螺旋波、外滚螺旋波、旅行螺旋波和内滚螺旋波的显著变化. 通过定义序参量来定量的描述由时空调制引起的螺旋波在不同态之间非平衡跃迁的临界条件, 及漫游螺旋波波头圆滚圆半径随调制参数的变化情况. 当时空调制增大到某个临界值时, 螺旋波发生了破碎; 再增加时空调制, 螺旋波则发生了衰减, 系统最终演化为空间均匀静息态. 在文中给出了螺旋波发生破碎和衰减的机理和原因. 最后将时空调制方法运用于漫游螺旋波, 实现了将漫游螺旋波控制成周期螺旋波, 或将其控制为空间均匀静息态.     

Spiral wave dynamics in excitable media with spherical geometries

We describe the spatial and temporal organization of spiral and scroll waves in spherical shells of different sizes and solid spheres. We present simulation results for the evolution of the dynamics and clustering of spiral waves as a function of the excitability of the medium. The excitability, topology, and size of the domain places restrictions on how single and multiarmed spiral waves are organized in space. The results in spherical geometries are compared with those in planar two-dimensional media. These studies are relevant to the dynamics of spiral waves in a variety of media including the heart, and chemical reactions on spherical surfaces.     

Wave front fragmentation due to ventricular geometry in a model of the rabbit heart

The role of the heart's complex shape in causing the fragmentation of activation wave fronts characteristic of ventricular fibrillation (VF) has not been well studied. We used a finite element model of cardiac propagation capable of simulating functional reentry on curved two-dimensional surfaces to test the hypothesis that uneven surface curvature can cause local propagation block leading to proliferation of reentrant wave fronts. We found that when reentry was induced on a flat sheet, it rotated in a repeatable meander pattern without breaking up. However, when a model of the rabbit ventricles was formed from the same medium, reentrant wave fronts followed complex, nonrepeating trajectories. Local propagation block often occurred when wave fronts propagated across regions where the Gaussian curvature of the surface changed rapidly. This type of block did not occur every time wave fronts crossed such a region; rather, it only occurred when the wave front was very close behind the previous wave in the cycle and was therefore propagating into relatively inexcitable tissue. Close wave front spacing resulted from nonstationary reentrant propagation. Thus, uneven surface curvature and nonstationary reentrant propagation worked in concert to produce wave front fragmentation and complex activation patterns. None of the factors previously thought to be necessary for local propagation block (e.g., heterogeneous refractory period, steep action potential duration restitution) were present. We conclude that the complex geometry of the heart may be an important determinant of VF activation patterns. (c) 2002 American Institute of Physics.     

Spiral wave generation in heterogeneous excitable media

As the coupling in a heterogeneous excitable medium is reduced, three different types of behavior are encountered: plane waves propagate without breaking up, plane waves break up into spiral waves, and plane waves block. We illustrate these phenomena in monolayers of chick embryonic heart cells using calcium sensitive fluorescent dyes. Following the addition of heptanol, an agent that reduces the electrical coupling between cells, we observe breakup of spiral waves. These results are modeled in a heterogeneous cellular automaton model in which the neighborhood of interaction is modified.     

Dynamics of Spiral Wave Tip in Excitable Media with Gradient Parameter

Using a Barkley model as an example, we study spiral waves and spiral tips in a gradient excitable medium. The gradient distribution of parameters is introduced to depict the inhomogeneous medium. It is found that the parameter fluctuations play an important role in the morphology of spiral pattern and the movements of spiral tips. For varied gradient parameters, it is observed that there exist three kinds of spiral behaviors, stable rotation, rebound of spiral tip from the boundary, and spiral breakup.     

Controlling Spiral Dynamics in Excitable Media by a Weakly Localized Pacing

Spiral dynamics controlled by a weakly localized pacing around the spiral tip is investigated. Numerical simulations show two distinct characteristics when the pacing is applied with the weak amplitude for suitable frequencies： for a rigidly .rotating spiral, a transition from rigid rotation to meandering motion is observed, and for unstable spiral waves, spiral breakup can be prevented. Successfully preventing spiral breakup is relevant to the modulation of the tip trajectory induced by a localized pacing.     

Tracking waves and vortex nucleation in excitable systems with anomalous dispersion

We report experimental results obtained from a chemical reaction-diffusion system in which wave propagation is limited to a finite band of wavelengths and in which no solitary pulses exist. Wave patterns increase their size through repeated annihilation events of the frontier pulse that allow the succeeding pulses to advance farther. A related type of wave dynamics involves a stable but slow frontier pulse that annihilates subsequent waves in front-to-back collisions. These so-called merging dynamics give rise to an unexpected form of spiral wave nucleation. All of these phenomena are reproduced by a simple, three-species reaction-diffusion model that reveals the importance of the underlying anomalous dispersion relation.     

From local to global spatiotemporal chaos in a cardiac tissue model

Two kinds of chaos can occur in cardiac tissue, chaotic meander of a single intact spiral wave and chaotic spiral wave breakup. We studied these behaviors in a model of two-dimensional cardiac tissue based on the Luo-Rudy I action potential model. In the chaotic meander regime, chaos is spatially localized to the core of the spiral wave. When persistent spiral wave breakup occurs, there is a transition from local to global spatiotemporal chaos.     

Development and transition of spiral wave in the coupled Hindmarsh--Rose neurons in two-dimensional space

The dynamics and the transition of spiral waves in the coupled Hindmarsh--Rose (H--R) neurons in two-dimensional space are investigated in the paper. It is found that the spiral wave can be induced and developed in the coupled HR neurons in two-dimensional space, with appropriate initial values and a parameter region given. However, the spiral wave could encounter instability when the intensity of the external current reaches a threshold value of 1.945. The transition of spiral wave is found to be affected by coupling intensity D and bifurcation parameter r. The spiral wave becomes sparse as the coupling intensity increases, while the spiral wave is eliminated and the whole neuronal system becomes homogeneous as the bifurcation parameter increases to a certain threshold value. Then the coupling action of the four sub-adjacent neurons, which is described by coupling coefficient D’, is also considered, and it is found that the spiral wave begins to breakup due to the introduced coupling action from the sub-adjacent neurons (or sites) and together with the coupling action of the nearest-neighbour neurons, which is described by the coupling intensity D.     

Conduction block in one-dimensional heart fibers

We present a nonlinear dynamical systems analysis of the transition to conduction block in one-dimensional cardiac fibers. We study a simple model of wave propagation in heart tissue that depends only on the recovery of action potential duration and conduction velocity. If the recovery function has slope >or=1 and the velocity recovery function is nonconstant, rapid activation causes dynamical heterogeneity and finally conduction block away from the activation site. This dynamical mechanism may play a role in the initiation and breakup of spiral waves in excitable media.     

Kinetic Monte Carlo simulations of travelling pulses and spiral waves in the lattice Lotka-Volterra model

Kinetic Monte Carlo simulations are used to study the stochastic two-species Lotka-Volterra model on a square lattice. For certain values of the model parameters, the system constitutes an excitable medium: travelling pulses and rotating spiral waves can be excited. Stable solitary pulses travel with constant (modulo stochastic fluctuations) shape and speed along a periodic lattice. The spiral waves observed persist sometimes for hundreds of rotations, but they are ultimately unstable and break-up (because of fluctuations and interactions between neighboring fronts) giving rise to complex dynamic behavior in which numerous small spiral waves rotate and interact with each other. It is interesting that travelling pulses and spiral waves can be exhibited by the model even for completely immobile species, due to the non-local reaction kinetics.